Maximal subgroups of free idempotent-generated semigroups factored out by biorder relations

David Easdown, Sean Gardiner and Brett McElwee

Abstract

Given any biordered set \(E\), we may form the idempotent-generated semigroup \(F_E\), which is generated by the set \(E\), subject to the relations \(ef=e*f\) whenever \(e\) and \(f\) are elements of \(E\) and \(e*f\) is a basic product. Easdown proved in 1985 that the biordered set of \(F_E\) is biorder isomorphic to \(E\), thus demonstrating that the biordered set axioms, introduced by Nambooripad in 1974, characterise certain partial algebras of idempotents of semigroups. Relatively little is known about the general structure of \(F_E\), though it is known that every group can arise as a maximal subgroup of \(F_E\) for some \(E\), and that, as a consequence, the word problem is unsolvable. In this article, presentations for maximal subgroups are studied using homomorphic images of fundamental groups of graphs associated with \({\mathcal D}\)-classes of the biordered set \(E\). To illustrate the technique, small biordered sets \(E\) are constructed where \(F_E\) contains maximal subgroups which are cyclic of order two and free abelian on two generators respectively, the second of which reconstructs an example due to Dolinka.

Keywords: free idempotent-generated semigroups, biordered sets, maximal subgroups.

AMS Subject Classification: Primary 06A99; secondary 20M10, 20M20.